That doesn't tell me what it is at all. Further, I don't see what closed form formulas have to do with infinite sums.

A closed-form formula is when you take a formula with an infinite sum, such as
[tex]
s = \sum_{k=0}^\infty ar^k
[/tex]
and simplify it to an algebraic formula, which in this case would be
[tex]
s = \frac{a}{1 - r}
[/tex]
(Assuming, in this case, that r < 1.)
Understand?

Perhaps the real question is, what do you want the closed form for? If you want to calculate values exactly, use the recurrence formula. To approximate it, use the asymptotic formula. But I assume you need the closed form for some other reason.

A closed-form formula is when you take a formula with an infinite sum, such as
[tex]
s = \sum_{k=0}^\infty ar^k
[/tex]
and simplify it to an algebraic formula, which in this case would be
[tex]
s = \frac{a}{1 - r}
[/tex]
(Assuming, in this case, that r < 1.)
Understand?

Couldnt we define a closed formula p(n) over a set of functions (for example the elementary functions, or maybe only the functions f(x)=x, f(x)=c) combined with a certain set of operations (for example ^*/-+) as a formula whose number of terms (functions):
1) is not infinite
2) is not depending on n.

Couldnt we define a closed formula p(n) over a set of functions (for example the elementary functions, or maybe only the functions f(x)=x, f(x)=c) combined with a certain set of operations (for example ^*/-+) as a formula whose number of terms (functions):
1) is not infinite
2) is not depending on n.

I'd prefer to define it symbolically, perhaps with a context-free grammar: